The percolation transition in the zero-temperature Domany model

We analyze a deterministic cellular automaton sigma=(sigma(n): ngreater than or equal to0) corresponding to the zero-temperature case of Domany's stochastic Ising ferromagnet on the hexagonal lattice H. The state space L-H={- 1,+ 1}(H) consists of assignments of - 1 or + 1 to each site of H and the initial state sigma(0)={sigma(x)(o)}(x is an element of H) is chosen randomly with P(sigma(x)(o) = +1)= p is an element of [0, 1]. The sites of H are partitioned in two sets A and B so that all the neighbors of a site x in A belong to B and vice versa, and the discrete time dynamics is such that the sigma(x)'s with x is an element of A ( respectively, B) are updated simultaneously at odd (resp., even) times, making sigma(x) agree with the majority of its three neighbors. In ref. 1 it was proved that there is a percolation transition at p = 1/2 in the percolation models defined by sigma(n), for all times n is an element of [1, infinity]. In this paper, we study the nature of that transition and prove that the critical exponents beta, nu, and eta of the dependent percolation models defined by sigma(n), n is an element of [1, infinity], have the same values as for standard two-dimensional independent site percolation ( on the triangular lattice).